The area of the (bounded) region between the curves y = x^3 - x + 3 and y = x^2 - x + 3 is a/12 , where a is = ?

Answer 1

The first thing we need to know is where the two functions intersect so that we know the limits of integration.

To find the intersection of two functions, we set them equal to each other,

x³ - x + 3 = x² - x + 3

We get some nice cancellation, leaving x³ = x². This is true only when x = 1 or x = 0, so those are the bounds of integration.

Next, we should figure out which function is larger. There are two ways to do this:

1. Plugging in a number in the range 0 to 1. Choosing x = 1/2,

(1/2)³ - (1/2) + 3 = 2.625

(1/2)² - (1/2) + 3 = 2.75

1. Looking at the functions. The only difference between them is that first term. In the range 0 to 1, x³ is smaller than x², so the first function must be smaller in that interval.

Now we need to integrate to find the area in between the graphs. If the larger function is f and the smaller function is g, then the area between them from b to c is

∫_b^c (f(x) - g(x)) dx

Plugging in what we know, this is

∫₀¹ ((x² - x + 3) - (x³ - x + 3)) dx which simplifies to

∫₀¹ (x² - x³) dx

x³ / 3 - x⁴ / 4 |₀¹

1/3 - 1/4

1/12

The original question said that the area is a/12, and we found that the area is 1/12, therefore the value of a must be 1.

Hope this helps!

~Logan